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Register a new userA Horrocks theorem for reflexive sheaves
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In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any $m$-tail reflexive sheaf $mathcal{E}$ on $mathcal{P}^n$ is greater or equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on $mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail reflexive sheaves of higher rank.
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We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $mathbb{P}^3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $mathscr{G}$ is a subfoliation of a codimension one distribution $mathscr{F}$ with isolated singularities, then $Sing(mathscr{G})$ is a curve. As a consequence, we give a criterion to decide whether $mathscr{G}$ is globally given as the intersection of $mathscr{F}$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.
An important classification problem in Algebraic Geometry deals with pairs $(E,phi)$, consisting of a torsion free sheaf $E$ and a non-trivial homomorphism $phicolon (E^{otimes a})^{oplus b}lradet(E)^{otimes c}otimes L$ on a polarized complex projective manifold $(X,O_X(1))$, the input data $a$, $b$, $c$, $L$ as well as the Hilbert polynomial of $E$ being fixed. The solution to the classification problem consists of a family of moduli spaces ${cal M}^delta:={cal M}^{delta-rm ss}_{a/b/c/L/P}$ for the $delta$-semistable objects, where $deltainQ[x]$ can be any positive polynomial of degree at most $dim X-1$. In this note we show that there are only finitely many distinct moduli spaces among the ${cal M}^delta$ and that they sit in a chain of GIT-flips. This property has been known and proved by ad hoc arguments in several special cases. In our paper, we apply refined information on the instability flag to solve this problem. This strategy is inspired by the fundamental paper of Ramanan and Ramanathan on the instability flag.
We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing $t$-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular, we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict ourselves to the two-strata case, but our results extend to the general case.
Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_Motimes_RM is zero. When R is a quasi-specialization of a G-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization of a G-regular local ring or a quasi-Buchsbaum local ring.
We present a nearby cycle sheaf construction in the context of symmetric spaces. This construction can be regarded as a replacement for the Grothendieck-Springer resolution in classical Springer theory.
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